Quantification In logic, quantification is the binding of a variable ranging over a domain of discourse. The variable thereby becomes bound by an operator called a quantifier. Academic discussion of quantification refers more often to this meaning of the term than the preceding one. Natural language[edit] All known human languages make use of quantification (Wiese 2004). Every glass in my recent order was chipped.Some of the people standing across the river have white armbands.Most of the people I talked to didn't have a clue who the candidates were.A lot of people are smart. The words in italics are quantifiers. The study of quantification in natural languages is much more difficult than the corresponding problem for formal languages. Montague grammar gives a novel formal semantics of natural languages. Logic[edit] In language and logic, quantification is a construct that specifies the quantity of specimens in the domain of discourse that apply to (or satisfy) an open formula. Mathematics[edit] . means

Model theory This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang and Keisler (1990):[1] universal algebra + logic = model theory. Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997): although model theorists are also interested in the study of fields. In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. Branches of model theory[edit] This article focuses on finitary first order model theory of infinite structures. During the last several decades applied model theory has repeatedly merged with the more pure stability theory. and or

Signature (logic) In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. Signatures play the same role in mathematics as type signatures in computer programming. They are rarely made explicit in more philosophical treatments of logic. Formally, a (single-sorted) signature can be defined as a triple σ = (Sfunc, Srel, ar), where Sfunc and Srel are disjoint sets not containing any other basic logical symbols, called respectively function symbols (examples: +, ×, 0, 1) andrelation symbols or predicates (examples: ≤, ∈), and a function ar: Sfunc Srel → which assigns a non-negative integer called arity to every function or relation symbol. A signature with no function symbols is called a relational signature, and a signature with no relation symbols is called an algebraic signature. Symbol types S.

Interpretation From Wikipedia, the free encyclopedia Interpretation may refer to: Culture[edit] Aesthetic interpretation, an explanation of the meaning of a work of artAllegorical interpretation, an approach that assumes a text should not be interpreted literallyDramatic Interpretation, an event in speech and forensics competitions in which participants perform excerpts from playsHeritage interpretation, communication about the nature and purpose of historical, natural, or cultural phenomenaInterpretation (music), the process of a performer deciding how to perform music that has been previously composedLanguage interpretation, the facilitation of dialogue between parties using different languagesLiterary theory, broad methods for interpreting literature, including historicism, feminism, structuralism, deconstruction Literary criticism, interpretation of particular works of literatureOral interpretation, a dramatic art Law[edit] Math and computing[edit] Media[edit] Neuroscience[edit] Philosophy[edit]

Cartesian product Cartesian product of the sets and The simplest case of a Cartesian product is the Cartesian square, which returns a set from two sets. A Cartesian product of n sets can be represented by an array of n dimensions, where each element is an n-tuple. The Cartesian product is named after René Descartes,[1] whose formulation of analytic geometry gave rise to the concept. Examples[edit] A deck of cards[edit] An illustrative example is the standard 52-card deck. Ranks × Suits returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}. Suits × Ranks returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}. A two-dimensional coordinate system[edit] An example in analytic geometry is the Cartesian plane. Most common implementation (set theory)[edit] A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. . , where For example: Similarly

Hydrogen Chemical element with symbol H and atomic number 1; lightest and most abundant chemical substance in the universe Chemical element, symbol H and atomic number 1 Hydrogen is nonmetallic, except at extremely high pressures, and readily forms a single covalent bond with most nonmetallic elements, forming compounds such as water and nearly all organic compounds. Hydrogen gas was first artificially produced in the early 16th century by the reaction of acids on metals. Industrial production is mainly from steam reforming natural gas, and less often from more energy-intensive methods such as the electrolysis of water.[12] Most hydrogen is used near the site of its production, the two largest uses being fossil fuel processing (e.g., hydrocracking) and ammonia production, mostly for the fertilizer market. Properties Combustion Combustion of hydrogen with the oxygen in the air. Hydrogen gas (dihydrogen or molecular hydrogen)[15] is highly flammable: The enthalpy of combustion is −286 kJ/mol.[16] Flame

Domain of discourse The term universe of discourse generally refers to the collection of objects being discussed in a specific discourse. In model-theoretical semantics, a universe of discourse is the set of entities that a model is based on. The concept universe of discourse is generally attributed to Augustus De Morgan (1846) but the name was used for the first time in history by George Boole (1854) on page 42 of his Laws of Thought in a long and incisive passage well worth study. Boole's definition is quoted below. The concept, probably discovered independently by Boole in 1847, played a crucial role in his philosophy of logic especially in his stunning principle of wholistic reference. A database is a model of some aspect of the reality of an organisation. Boole’s 1854 Definition[edit] See also[edit] References[edit] Jump up ^ Corcoran, John.

Arity In logic, mathematics, and computer science, the arity Examples[edit] The term "arity" is rarely employed in everyday usage. A nullary function takes no arguments.A unary function takes one argument.A binary function takes two arguments.A ternary function takes three arguments.An n-ary function takes n arguments. Nullary[edit] Unary[edit] Binary[edit] Most operators encountered in programming are of the binary form. Ternary[edit] with arbitrary precision. n-ary[edit] From a mathematical point of view, a function of n arguments can always be considered as a function of one single argument which is an element of some product space. The same is true for programming languages, where functions taking several arguments could always be defined as functions taking a single argument of some composite type such as a tuple, or in languages with higher-order functions, by currying. Variable arity[edit] In computer science, a function accepting a variable number of arguments is called variadic.

What is a homunculus and what does it tell scientists A homunculus is a sensory map of your body, so it looks like an oddly proportioned human. The reason it's oddly proportioned is that a homunculus represents each part of the body in proportion to its number of sensory neural connections and not its actual size. The layout of the sensory neural connections throughout your body determines the level of sensitivity each area of your body has, so the hands on a sensory homunculus are its largest body parts, exaggerated to an almost comical degree, while the arms are quite skinny. The homunculus, then, gives a vivid picture of where our sensory system gets the most bang for its buck. Chemoreceptors, which sense chemicals.

Analytic–synthetic distinction Semantic distinction in philosophy While the distinction was first proposed by Immanuel Kant, it was revised considerably over time, and different philosophers have used the terms in very different ways. Furthermore, some philosophers (starting with Willard Van Orman Quine) have questioned whether there is even a clear distinction to be made between propositions which are analytically true and propositions which are synthetically true.[2] Debates regarding the nature and usefulness of the distinction continue to this day in contemporary philosophy of language.[2] Conceptual containment [edit] The philosopher Immanuel Kant uses the terms "analytic" and "synthetic" to divide propositions into two types. analytic proposition: a proposition whose predicate concept is contained in its subject conceptsynthetic proposition: a proposition whose predicate concept is not contained in its subject concept but related Examples of analytic propositions, on Kant's definition, include: Logical positivists

Well-formed formula Introduction[edit] A key use of formulae is in propositional logic and predicate logics such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven. Although the term "formula" may be used for written marks (for instance, on a piece of paper or chalkboard), it is more precisely understood as the sequence being expressed, with the marks being a token instance of formula. Propositional calculus[edit] The formulas of propositional calculus, also called propositional formulas,[2] are expressions such as . The formulae are inductively defined as follows: Each propositional variable is, on its own, a formula.If φ is a formula, then φ is a formula.If φ and ψ are formulas, and • is any binary connective, then ( φ • ψ) is a formula. q)

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